Maths - Predicate Category

See also topos theory.

The relationship X contains I (X is a subset of I) is a predicate. We can form a category of predicates on sets as follows:

Objects

predicates are pairs (I,X) where
X contains I

X(i) implies an element i∈I is a free variable in I.

predicate category 1
Morphisms

(I,X) -> (J,Y) where:

u:I-> J and X(i) implies: Y(u(i))

So, to define(I,X) -> (J,Y) , we need maps u:I -> J and X -> Y but X -> Y is implied by u.

predicate category

Reverse map

'subsitution functor' u*

examples of substitution are:

  • weakening
  • contraction
reverse
weakening   weakening
contraction   contraction
quantifiers   quantifier

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